Question

Using the the concept of rank of the matrix find whether the following system of equations

are consistent or inconsistent,

i.

4y + z = 0

12x - 5y - 3z = 34

-6x + 4z = 8

Answer 1

i. $Ax=b$ is inconsistent (i.e., no solution exists) if and only if $\text{rank}[A]<\text{rank}[A|b].$

ii. $Ax=b$ has an unique solution if and only if $\text{rank}[A]=\text{rank}[A|b]=n.$

iii. $Ax=b$ has infinitely many solutions if and only if $\text{rank}[A]=\text{rank}[A|b]<n.$

$$A=\begin{bmatrix} 0 & 4 & 1 \\ 12 & -5 & -3 \\ -6 & 0 & 4 \end{bmatrix}$$

Swap rows 1and 2

$$\begin{bmatrix} 12 & -5 & -3 \\ 0 & 4 & 1 \\ -6 & 0 & 4 \end{bmatrix}$$

$R_3=R_3+(1/2)R_1$

$$\begin{bmatrix} 12 & -5 & -3 \\ 0 & 4 & 1 \\ 0 & -5/2 & 5/2 \end{bmatrix}$$

$R_3=R_3+(5/8)R_2$

The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is 3.

Since $\text{rank}[A]=\text{rank}[A|b]=3=n,$ then $Ax=b$ has an unique solution.

Therefore, the given system of equations is consistent.